Question 5.10: Characterize the composite waveform generated by subtracting......
Characterize the composite waveform generated by subtracting an exponential from a step function with the same amplitude.
The equation for this composite waveform is
υ(t) = V_{A}u(t) − [V_{A}e^{−t/T_{C}}] u(t) V
= V_{A}[1 – e^{−t/T_{C}}] u(t) V
For t < 0 the waveform is zero because of the step function. At t = 0 the waveform is still zero since the step and exponential cancel.
υ(0) = V_{A}[1 – e^{0}] (1) = 0
For t \gg T_{C} the waveform approaches a constant value VA because the exponential term decays to zero. For practical purposes υ(t) is within less than 1% of its final value VA when t = 5TC. At t = TC, υ(TC) = VA (1 − e-1) = 0.632 VA. The waveform rises to about 63% of its final value in one time constant. All of the observations are summarized in the plot shown in Figure 5–28. This waveform is called an exponential rise. It is also sometimes referred to as a “charging exponential,” since it represents the behavior of signals that occur during the buildup of voltage in resistor-capacitor circuits studied in Chapter 7.
